Drow the graph of power function y=x⅔
Show the table value
Domain =?
Range=?
To plot the graph of the power function y = x^(2/3), we can start by determining some key points and then connecting them to form a curve.
Table of values:
x = -8: y = (-8)^(2/3) ≈ 3.17
x = -4: y = (-4)^(2/3) ≈ 2
x = -2: y = (-2)^(2/3) ≈ 1.59
x = -1: y = (-1)^(2/3) = 1
x = 0: y = 0^(2/3) = 0
x = 1: y = 1^(2/3) = 1
x = 2: y = 2^(2/3) ≈ 1.59
x = 4: y = 4^(2/3) ≈ 2
x = 8: y = 8^(2/3) ≈ 3.17
Domain: The function y = x^(2/3) is defined for all real numbers.
Range: The range of the function depends on the domain. However, the range of y = x^(2/3) for all real numbers is the set of non-negative real numbers.
Graph:
Let's plot the points and connect them to visualize the graph.
Here is the graph of the power function y = x^(2/3):
```
|
3.17| x
|
|
2 |
|
|
1.59 |
|
1 | x
|
|
0 |
-------------------|-----------------------
-8 -4 -2 -1 0 1 2 4 8
```
Please note that the graph does not include negative values for the x-axis because the even-root function is only defined for non-negative real numbers.
To draw the graph of the power function y = x^(2/3), we can start by plotting some points. We'll choose values of x and calculate the corresponding y values.
Let's pick a few values for x, say -8, -4, -2, -1, 0, 1, 2, 4, and 8.
For x = -8: y = (-8)^(2/3) ≈ 4.66
For x = -4: y = (-4)^(2/3) ≈ 2.83
For x = -2: y = (-2)^(2/3) ≈ 1.59
For x = -1: y = (-1)^(2/3) ≈ 1
For x = 0: y = 0^(2/3) = 0
For x = 1: y = 1^(2/3) = 1
For x = 2: y = 2^(2/3) ≈ 1.59
For x = 4: y = 4^(2/3) ≈ 2.83
For x = 8: y = 8^(2/3) ≈ 4.66
Now we can plot these points on a graph:
|
5 | ○
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4 | ○
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3 |
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2 | ○
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1 | ○
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0 | ○
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-1 |
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-2 | ○
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-3 |
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-4 | ○
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-5 |
|
-----------------------------
-8 -6 -4 -2 0 2 4 6 8
This graph is a curve that starts from the origin (0,0) and curves upwards to the right. It is symmetric about the y-axis. The graph never crosses the x-axis because there is no real value of x that gives a negative value of y.
Now, let's examine the domain and range of the power function y = x^(2/3).
The domain of the function is the set of all real numbers because any real number can be raised to the power of \frac{2}{3}.
The range of the function is all non-negative real numbers, including zero. This is because raising a real number to the power of \frac{2}{3} always results in a non-negative value, or zero if x is zero.
Hope this helps!
To draw the graph of the power function y = x^(2/3), we can start by plotting a few points and then connecting them to visualize the shape of the curve.
Table of values:
To create a table of values, we can choose different x-values and determine the corresponding y-values using the given function.
Let's select some x-values and find their corresponding y-values:
For simplicity, let's consider x-values from -8 to 8.
x | y
-------------
-8 | (-8)^(2/3)
-4 | (-4)^(2/3)
-2 | (-2)^(2/3)
0 | 0^(2/3)
2 | 2^(2/3)
4 | 4^(2/3)
8 | 8^(2/3)
Calculate the corresponding y-values by evaluating the function for each x-value.
The domain represents all possible x-values for which the function is defined. In this case, since x can be any real number, the domain is (-∞, ∞), which means the function is defined for all real numbers.
The range represents all possible y-values or outputs of the function. For this power function, since x is raised to the power of (2/3), the range will include all non-negative real numbers. Therefore, the range is [0, ∞).
Now, we can plot the points obtained and connect them to create the graph. Each point's x-coordinate represents the x-value chosen, and its y-coordinate represents the corresponding y-value.
After plotting all the points and connecting them, you will have the graph of the power function y = x^(2/3).